LMFDB - Embedded newform 2700.2.i.d.901.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(901,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.901"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-1,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.142635249.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 $ x^8 - 3x^7 + 3x^6 + 3x^5 - 11x^4 + 6x^3 + 12x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			901.1
Root			$0.818235 - 1.15347i$ of defining polynomial
Character	$\chi$	$=$	2700.901
Dual form			2700.2.i.d.1801.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.49787 - 4.32643i) q^{7} +(1.99787 + 3.46041i) q^{11} +(0.771582 - 1.33642i) q^{13} -6.99574 q^{17} -2.25667 q^{19} +(-3.89778 + 6.75116i) q^{23} +(3.08304 + 5.33998i) q^{29} +(0.271582 - 0.470394i) q^{31} +6.25240 q^{37} +(0.0979532 - 0.169660i) q^{41} +(-0.0431636 - 0.0747616i) q^{43} +(1.91483 + 3.31658i) q^{47} +(-8.97869 + 15.5515i) q^{49} +4.19164 q^{53} +(3.51278 - 6.08432i) q^{59} +(1.45470 + 2.51962i) q^{61} +(4.48295 - 7.76470i) q^{67} -8.79130 q^{71} +2.28650 q^{73} +(9.98082 - 17.2873i) q^{77} +(6.32211 + 10.9502i) q^{79} +(6.98082 + 12.0911i) q^{83} +10.3577 q^{89} -7.70924 q^{91} +(4.66936 + 8.08758i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{7} - 3 q^{11} + 2 q^{13} - 18 q^{17} - 8 q^{19} - 3 q^{23} + 9 q^{29} - 2 q^{31} + 2 q^{37} - 9 q^{41} + 8 q^{43} + 12 q^{47} - 9 q^{49} - 24 q^{53} + 15 q^{59} + q^{61} + 11 q^{67} + 24 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) $ 8 * q - q^7 - 3 * q^11 + 2 * q^13 - 18 * q^17 - 8 * q^19 - 3 * q^23 + 9 * q^29 - 2 * q^31 + 2 * q^37 - 9 * q^41 + 8 * q^43 + 12 * q^47 - 9 * q^49 - 24 * q^53 + 15 * q^59 + q^61 + 11 * q^67 + 24 * q^71 + 20 * q^73 + 36 * q^77 + 7 * q^79 + 12 * q^83 - 6 * q^89 - 22 * q^91 + 5 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.49787	−	4.32643i	−0.944105	−	1.63524i	−0.757533	−	0.652797i	$-0.773595\pi$
$7$	−2.49787	−	4.32643i	−0.944105	−	1.63524i	−0.186573	−	0.982441i	$-0.559738\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.99787	+	3.46041i	0.602380	+	1.04335i	0.992460	+	0.122571i	$0.0391141\pi$
$11$	1.99787	+	3.46041i	0.602380	+	1.04335i	−0.390080	+	0.920781i	$0.627553\pi$
$12$		0			0
$13$	0.771582	−	1.33642i	0.213998	−	0.370656i	−0.738964	−	0.673745i	$-0.764685\pi$
$13$	0.771582	−	1.33642i	0.213998	−	0.370656i	0.952962	+	0.303089i	$0.0980180\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.99574			−1.69672			−0.848358	−	0.529424i	$-0.822408\pi$
$17$	−6.99574			−1.69672			−0.848358	+	0.529424i	$0.822408\pi$
$18$		0			0
$19$	−2.25667			−0.517715			−0.258857	−	0.965916i	$-0.583346\pi$
$19$	−2.25667			−0.517715			−0.258857	+	0.965916i	$0.583346\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.89778	+	6.75116i	−0.812744	+	1.40771i	0.0981929	+	0.995167i	$0.468694\pi$
$23$	−3.89778	+	6.75116i	−0.812744	+	1.40771i	−0.910937	+	0.412546i	$0.864640\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.08304	+	5.33998i	0.572506	+	0.991609i	0.996308	+	0.0858540i	$0.0273619\pi$
$29$	3.08304	+	5.33998i	0.572506	+	0.991609i	−0.423802	+	0.905755i	$0.639305\pi$
$30$		0			0
$31$	0.271582	−	0.470394i	0.0487775	−	0.0844852i	−0.840606	−	0.541647i	$-0.817801\pi$
$31$	0.271582	−	0.470394i	0.0487775	−	0.0844852i	0.889383	+	0.457162i	$0.151134\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.25240			1.02789			0.513944	−	0.857824i	$-0.328184\pi$
$37$	6.25240			1.02789			0.513944	+	0.857824i	$0.328184\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.0979532	−	0.169660i	0.0152977	−	0.0264964i	−0.858275	−	0.513190i	$-0.828464\pi$
$41$	0.0979532	−	0.169660i	0.0152977	−	0.0264964i	0.873573	+	0.486693i	$0.161797\pi$
$42$		0			0
$43$	−0.0431636	−	0.0747616i	−0.00658239	−	0.0114010i	0.862715	−	0.505690i	$-0.168762\pi$
$43$	−0.0431636	−	0.0747616i	−0.00658239	−	0.0114010i	−0.869298	+	0.494289i	$0.835429\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.91483	+	3.31658i	0.279307	+	0.483774i	0.971213	−	0.238214i	$-0.0765620\pi$
$47$	1.91483	+	3.31658i	0.279307	+	0.483774i	−0.691906	+	0.721988i	$0.743229\pi$
$48$		0			0
$49$	−8.97869	+	15.5515i	−1.28267	+	2.22165i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.19164			0.575766			0.287883	−	0.957666i	$-0.407049\pi$
$53$	4.19164			0.575766			0.287883	+	0.957666i	$0.407049\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.51278	−	6.08432i	0.457326	−	0.792111i	−0.541493	−	0.840705i	$-0.682141\pi$
$59$	3.51278	−	6.08432i	0.457326	−	0.792111i	0.998819	+	0.0485942i	$0.0154741\pi$
$60$		0			0
$61$	1.45470	+	2.51962i	0.186256	+	0.322604i	0.943999	−	0.329948i	$-0.107031\pi$
$61$	1.45470	+	2.51962i	0.186256	+	0.322604i	−0.757743	+	0.652553i	$0.773698\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.48295	−	7.76470i	0.547680	−	0.948609i	−0.450753	−	0.892649i	$-0.648845\pi$
$67$	4.48295	−	7.76470i	0.547680	−	0.948609i	0.998433	−	0.0559605i	$-0.0178221\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.79130			−1.04334			−0.521668	−	0.853149i	$-0.674690\pi$
$71$	−8.79130			−1.04334			−0.521668	+	0.853149i	$0.674690\pi$
$72$		0			0
$73$	2.28650			0.267614			0.133807	−	0.991007i	$-0.457280\pi$
$73$	2.28650			0.267614			0.133807	+	0.991007i	$0.457280\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.98082	−	17.2873i	1.13742	−	1.97007i
$78$		0			0
$79$	6.32211	+	10.9502i	0.711293	+	1.23199i	0.964372	+	0.264549i	$0.0852232\pi$
$79$	6.32211	+	10.9502i	0.711293	+	1.23199i	−0.253080	+	0.967445i	$0.581443\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.98082	+	12.0911i	0.766244	+	1.32717i	0.939586	+	0.342313i	$0.111210\pi$
$83$	6.98082	+	12.0911i	0.766244	+	1.32717i	−0.173342	+	0.984862i	$0.555456\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.3577			1.09792			0.548958	−	0.835850i	$-0.315025\pi$
$89$	10.3577			1.09792			0.548958	+	0.835850i	$0.315025\pi$
$90$		0			0
$91$	−7.70924			−0.808148
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.66936	+	8.08758i	0.474102	+	0.821169i	0.999560	−	0.0296505i	$-0.00943942\pi$
$97$	4.66936	+	8.08758i	0.474102	+	0.821169i	−0.525458	+	0.850819i	$0.676106\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.98082	+	6.89498i	0.396106	+	0.686076i	0.993242	−	0.116064i	$-0.0370278\pi$
$101$	3.98082	+	6.89498i	0.396106	+	0.686076i	−0.597135	+	0.802140i	$0.703694\pi$
$102$		0			0
$103$	−8.20711	+	14.2151i	−0.808670	+	1.40066i	0.105115	+	0.994460i	$0.466479\pi$
$103$	−8.20711	+	14.2151i	−0.808670	+	1.40066i	−0.913785	+	0.406198i	$0.866854\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.3662			−1.09882			−0.549408	−	0.835554i	$-0.685147\pi$
$107$	−11.3662			−1.09882			−0.549408	+	0.835554i	$0.685147\pi$
$108$		0			0
$109$	−2.22683			−0.213292			−0.106646	−	0.994297i	$-0.534011\pi$
$109$	−2.22683			−0.213292			−0.106646	+	0.994297i	$0.534011\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.41696	−	2.45425i	0.133297	−	0.230877i	−0.791649	−	0.610976i	$-0.790777\pi$
$113$	1.41696	−	2.45425i	0.133297	−	0.230877i	0.924945	+	0.380100i	$0.124110\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	17.4744	+	30.2666i	1.60188	+	2.77453i
$120$		0			0
$121$	−2.48295	+	4.30060i	−0.225723	+	0.390963i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.02130			−0.623040			−0.311520	−	0.950240i	$-0.600838\pi$
$127$	−7.02130			−0.623040			−0.311520	+	0.950240i	$0.600838\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.18312	+	7.24538i	−0.365481	+	0.633032i	−0.988853	−	0.148893i	$-0.952429\pi$
$131$	−4.18312	+	7.24538i	−0.365481	+	0.633032i	0.623372	+	0.781925i	$0.285762\pi$
$132$		0			0
$133$	5.63685	+	9.76331i	0.488777	+	0.846587i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.39565	−	2.41734i	−0.119238	−	0.206527i	0.800228	−	0.599696i	$-0.204712\pi$
$137$	−1.39565	−	2.41734i	−0.119238	−	0.206527i	−0.919466	+	0.393169i	$0.871379\pi$
$138$		0			0
$139$	1.95799	−	3.39135i	0.166075	−	0.287650i	−0.770962	−	0.636882i	$-0.780224\pi$
$139$	1.95799	−	3.39135i	0.166075	−	0.287650i	0.937036	+	0.349231i	$0.113557\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.16607			0.515633
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.37860	+	11.0481i	−0.522555	+	0.905093i	0.477100	+	0.878849i	$0.341688\pi$
$149$	−6.37860	+	11.0481i	−0.522555	+	0.905093i	−0.999656	+	0.0262436i	$0.991645\pi$
$150$		0			0
$151$	0.188545	+	0.326569i	0.0153436	+	0.0265758i	0.873595	−	0.486653i	$-0.161782\pi$
$151$	0.188545	+	0.326569i	0.0153436	+	0.0265758i	−0.858252	+	0.513229i	$0.828449\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.04839	−	10.4761i	0.482714	−	0.836086i	−0.517089	−	0.855932i	$-0.672984\pi$
$157$	6.04839	−	10.4761i	0.482714	−	0.836086i	0.999803	+	0.0198461i	$0.00631762\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	38.9446			3.06926
$162$		0			0
$163$	−10.8563			−0.850333			−0.425166	−	0.905115i	$-0.639784\pi$
$163$	−10.8563			−0.850333			−0.425166	+	0.905115i	$0.639784\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.59369	+	4.49240i	−0.200706	+	0.347632i	−0.948756	−	0.316010i	$-0.897657\pi$
$167$	−2.59369	+	4.49240i	−0.200706	+	0.347632i	0.748050	+	0.663642i	$0.230990\pi$
$168$		0			0
$169$	5.30932	+	9.19602i	0.408409	+	0.707386i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.68739	+	2.92264i	0.128290	+	0.222204i	0.923014	−	0.384766i	$-0.125718\pi$
$173$	1.68739	+	2.92264i	0.128290	+	0.222204i	−0.794724	+	0.606970i	$0.792385\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.5997			−1.46495			−0.732474	−	0.680795i	$-0.761634\pi$
$179$	−19.5997			−1.46495			−0.732474	+	0.680795i	$0.761634\pi$
$180$		0			0
$181$	5.84865			0.434727			0.217363	−	0.976091i	$-0.430254\pi$
$181$	5.84865			0.434727			0.217363	+	0.976091i	$0.430254\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−13.9766	−	24.2081i	−1.02207	−	1.77027i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.2956	−	21.2965i	−0.889676	−	1.54096i	−0.840259	−	0.542186i	$-0.817597\pi$
$191$	−12.2956	−	21.2965i	−0.889676	−	1.54096i	−0.0494171	−	0.998778i	$-0.515736\pi$
$192$		0			0
$193$	−5.17150	+	8.95729i	−0.372252	+	0.644760i	−0.989912	−	0.141686i	$-0.954748\pi$
$193$	−5.17150	+	8.95729i	−0.372252	+	0.644760i	0.617659	+	0.786446i	$0.288081\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−16.0256			−1.14177			−0.570887	−	0.821028i	$-0.693401\pi$
$197$	−16.0256			−1.14177			−0.570887	+	0.821028i	$0.693401\pi$
$198$		0			0
$199$	−22.2353			−1.57622			−0.788111	−	0.615533i	$-0.788941\pi$
$199$	−22.2353			−1.57622			−0.788111	+	0.615533i	$0.788941\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.4020	−	26.6771i	1.08101	−	1.87237i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.50852	−	7.80899i	−0.311861	−	0.540159i
$210$		0			0
$211$	−12.1779	+	21.0927i	−0.838360	+	1.45208i	0.0529050	+	0.998600i	$0.483152\pi$
$211$	−12.1779	+	21.0927i	−0.838360	+	1.45208i	−0.891265	+	0.453483i	$0.850181\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.71350			−0.184205
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.39778	+	9.34923i	−0.363094	+	0.628897i
$222$		0			0
$223$	5.62407	+	9.74117i	0.376615	+	0.652317i	0.990567	−	0.137026i	$-0.0437545\pi$
$223$	5.62407	+	9.74117i	0.376615	+	0.652317i	−0.613952	+	0.789343i	$0.710421\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.07451	−	8.78931i	−0.336807	−	0.583367i	0.647023	−	0.762470i	$-0.276014\pi$
$227$	−5.07451	−	8.78931i	−0.336807	−	0.583367i	−0.983830	+	0.179103i	$0.942680\pi$
$228$		0			0
$229$	1.18428	−	2.05123i	0.0782595	−	0.135549i	−0.824240	−	0.566241i	$-0.808397\pi$
$229$	1.18428	−	2.05123i	0.0782595	−	0.135549i	0.902499	+	0.430692i	$0.141730\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−24.1831			−1.58429			−0.792144	−	0.610334i	$-0.791035\pi$
$233$	−24.1831			−1.58429			−0.792144	+	0.610334i	$0.791035\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.897782	+	1.55500i	−0.0580727	+	0.100585i	−0.893600	−	0.448864i	$-0.851829\pi$
$239$	−0.897782	+	1.55500i	−0.0580727	+	0.100585i	0.835528	+	0.549449i	$0.185162\pi$
$240$		0			0
$241$	7.43339	+	12.8750i	0.478827	+	0.829352i	0.999705	−	0.0242785i	$-0.00772884\pi$
$241$	7.43339	+	12.8750i	0.478827	+	0.829352i	−0.520878	+	0.853631i	$0.674396\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.74120	+	3.01585i	−0.110790	+	0.191894i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.8297			0.999159			0.499580	−	0.866268i	$-0.333488\pi$
$251$	15.8297			0.999159			0.499580	+	0.866268i	$0.333488\pi$
$252$		0			0
$253$	−31.1490			−1.95832
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.9979	−	19.0489i	0.686028	−	1.18824i	−0.287084	−	0.957905i	$-0.592686\pi$
$257$	10.9979	−	19.0489i	0.686028	−	1.18824i	0.973113	−	0.230330i	$-0.0739807\pi$
$258$		0			0
$259$	−15.6177	−	27.0506i	−0.970435	−	1.68084i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.57664	+	13.1231i	0.467196	+	0.809207i	0.999298	−	0.0374737i	$-0.0119310\pi$
$263$	7.57664	+	13.1231i	0.467196	+	0.809207i	−0.532102	+	0.846680i	$0.678598\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.99147			0.487249			0.243624	−	0.969870i	$-0.421664\pi$
$269$	7.99147			0.487249			0.243624	+	0.969870i	$0.421664\pi$
$270$		0			0
$271$	3.97248			0.241311			0.120656	−	0.992694i	$-0.461500\pi$
$271$	3.97248			0.241311			0.120656	+	0.992694i	$0.461500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	9.10702	+	15.7738i	0.547188	+	0.947757i	0.998466	+	0.0553736i	$0.0176350\pi$
$277$	9.10702	+	15.7738i	0.547188	+	0.947757i	−0.451278	+	0.892383i	$0.649032\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.1043	−	19.2333i	−0.662429	−	1.14736i	−0.979975	−	0.199119i	$-0.936192\pi$
$281$	−11.1043	−	19.2333i	−0.662429	−	1.14736i	0.317546	−	0.948243i	$-0.397141\pi$
$282$		0			0
$283$	5.82637	−	10.0916i	0.346342	−	0.599882i	−0.639255	−	0.768995i	$-0.720757\pi$
$283$	5.82637	−	10.0916i	0.346342	−	0.599882i	0.985597	+	0.169113i	$0.0540904\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.978696			−0.0577706
$288$		0			0
$289$	31.9403			1.87884
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.09156	+	5.35473i	−0.180611	+	0.312827i	−0.942089	−	0.335364i	$-0.891141\pi$
$293$	−3.09156	+	5.35473i	−0.180611	+	0.312827i	0.761478	+	0.648191i	$0.224474\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.01492	+	10.4181i	0.347852	+	0.602497i
$300$		0			0
$301$	−0.215634	+	0.373489i	−0.0124289	+	0.0215276i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.9872			0.741219			0.370610	−	0.928789i	$-0.379149\pi$
$307$	12.9872			0.741219			0.370610	+	0.928789i	$0.379149\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.0958	+	19.2185i	−0.629186	+	1.08978i	0.358529	+	0.933519i	$0.383278\pi$
$311$	−11.0958	+	19.2185i	−0.629186	+	1.08978i	−0.987715	+	0.156264i	$0.950055\pi$
$312$		0			0
$313$	−8.34609	−	14.4559i	−0.471749	−	0.817093i	0.527729	−	0.849413i	$-0.323044\pi$
$313$	−8.34609	−	14.4559i	−0.471749	−	0.817093i	−0.999478	+	0.0323199i	$0.989710\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.4787	+	28.5419i	0.925535	+	1.60307i	0.790698	+	0.612206i	$0.209718\pi$
$317$	16.4787	+	28.5419i	0.925535	+	1.60307i	0.134837	+	0.990868i	$0.456949\pi$
$318$		0			0
$319$	−12.3190	+	21.3371i	−0.689732	+	1.19465i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	15.7870			0.878414
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.56599	−	16.5688i	0.527390	−	0.913466i
$330$		0			0
$331$	−14.3333	−	24.8260i	−0.787830	−	1.36456i	−0.927294	−	0.374334i	$-0.877871\pi$
$331$	−14.3333	−	24.8260i	−0.787830	−	1.36456i	0.139464	−	0.990227i	$-0.455462\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.9270	+	25.8543i	−0.813125	+	1.40837i	0.0975411	+	0.995231i	$0.468902\pi$
$337$	−14.9270	+	25.8543i	−0.813125	+	1.40837i	−0.910666	+	0.413143i	$0.864431\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.17034			0.117530
$342$		0			0
$343$	54.7401			2.95569
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.41910	−	4.19000i	0.129864	−	0.224931i	−0.793760	−	0.608231i	$-0.791879\pi$
$347$	2.41910	−	4.19000i	0.129864	−	0.224931i	0.923624	+	0.383301i	$0.125213\pi$
$348$		0			0
$349$	16.0325	+	27.7691i	0.858200	+	1.48645i	0.873644	+	0.486566i	$0.161751\pi$
$349$	16.0325	+	27.7691i	0.858200	+	1.48645i	−0.0154438	+	0.999881i	$0.504916\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.92975	+	8.53857i	0.262384	+	0.454462i	0.966875	−	0.255251i	$-0.0821580\pi$
$353$	4.92975	+	8.53857i	0.262384	+	0.454462i	−0.704491	+	0.709713i	$0.748825\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.93266			−0.524226			−0.262113	−	0.965037i	$-0.584419\pi$
$359$	−9.93266			−0.524226			−0.262113	+	0.965037i	$0.584419\pi$
$360$		0			0
$361$	−13.9075			−0.731972
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	9.51065	+	16.4729i	0.496452	+	0.859880i	0.999992	−	0.00409206i	$-0.00130255\pi$
$367$	9.51065	+	16.4729i	0.496452	+	0.859880i	−0.503540	+	0.863972i	$0.667969\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10.4702	−	18.1349i	−0.543584	−	0.941515i
$372$		0			0
$373$	−0.939241	+	1.62681i	−0.0486320	+	0.0842332i	−0.889317	−	0.457292i	$-0.848820\pi$
$373$	−0.939241	+	1.62681i	−0.0486320	+	0.0842332i	0.840685	+	0.541525i	$0.182153\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.51526			0.490061
$378$		0			0
$379$	22.8435			1.17339			0.586697	−	0.809807i	$-0.300428\pi$
$379$	22.8435			1.17339			0.586697	+	0.809807i	$0.300428\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.35133	−	4.07262i	0.120147	−	0.208101i	−0.799678	−	0.600429i	$-0.794997\pi$
$383$	2.35133	−	4.07262i	0.120147	−	0.208101i	0.919826	+	0.392327i	$0.128330\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.38287	−	9.32340i	−0.272922	−	0.472715i	0.696687	−	0.717376i	$-0.254657\pi$
$389$	−5.38287	−	9.32340i	−0.272922	−	0.472715i	−0.969609	+	0.244661i	$0.921323\pi$
$390$		0			0
$391$	27.2679	−	47.2293i	1.37899	−	2.38849i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	8.10958			0.407008			0.203504	−	0.979074i	$-0.434767\pi$
$397$	8.10958			0.407008			0.203504	+	0.979074i	$0.434767\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.22744	+	10.7862i	−0.310984	+	0.538640i	−0.978576	−	0.205888i	$-0.933992\pi$
$401$	−6.22744	+	10.7862i	−0.310984	+	0.538640i	0.667592	+	0.744527i	$0.267325\pi$
$402$		0			0
$403$	−0.419095	−	0.725894i	−0.0208766	−	0.0361594i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.4915	+	21.6359i	0.619179	+	1.07245i
$408$		0			0
$409$	15.7166	−	27.2219i	0.777136	−	1.34604i	−0.156450	−	0.987686i	$-0.550005\pi$
$409$	15.7166	−	27.2219i	0.777136	−	1.34604i	0.933586	−	0.358353i	$-0.116662\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−35.0979			−1.72705
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.1848	−	22.8368i	0.644121	−	1.11565i	−0.340383	−	0.940287i	$-0.610557\pi$
$419$	13.1848	−	22.8368i	0.644121	−	1.11565i	0.984504	−	0.175363i	$-0.0561100\pi$
$420$		0			0
$421$	6.61555	+	11.4585i	0.322422	+	0.558451i	0.980987	−	0.194072i	$-0.0621696\pi$
$421$	6.61555	+	11.4585i	0.322422	+	0.558451i	−0.658565	+	0.752524i	$0.728836\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	7.26732	−	12.5874i	0.351690	−	0.609145i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−24.2895			−1.16999			−0.584993	−	0.811039i	$-0.698903\pi$
$431$	−24.2895			−1.16999			−0.584993	+	0.811039i	$0.698903\pi$
$432$		0			0
$433$	17.9840			0.864258			0.432129	−	0.901812i	$-0.357763\pi$
$433$	17.9840			0.864258			0.432129	+	0.901812i	$0.357763\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.79599	−	15.2351i	0.420769	−	0.728794i
$438$		0			0
$439$	−11.3577	−	19.6721i	−0.542074	−	0.938900i	−0.998785	−	0.0492847i	$-0.984306\pi$
$439$	−11.3577	−	19.6721i	−0.542074	−	0.938900i	0.456711	−	0.889615i	$-0.349028\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−11.4578	−	19.8455i	−0.544377	−	0.942888i	−0.998646	−	0.0520235i	$-0.983433\pi$
$443$	−11.4578	−	19.8455i	−0.544377	−	0.942888i	0.454269	−	0.890864i	$-0.349900\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.36966			0.253410			0.126705	−	0.991940i	$-0.459560\pi$
$449$	5.36966			0.253410			0.126705	+	0.991940i	$0.459560\pi$
$450$		0			0
$451$	0.782790			0.0368601
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−9.62309	−	16.6677i	−0.450149	−	0.779682i	0.548246	−	0.836317i	$-0.315296\pi$
$457$	−9.62309	−	16.6677i	−0.450149	−	0.779682i	−0.998395	+	0.0566358i	$0.981963\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.6870	+	30.6347i	0.823763	+	1.42680i	0.902861	+	0.429933i	$0.141463\pi$
$461$	17.6870	+	30.6347i	0.823763	+	1.42680i	−0.0790973	+	0.996867i	$0.525204\pi$
$462$		0			0
$463$	8.56075	−	14.8277i	0.397852	−	0.689100i	−0.595609	−	0.803275i	$-0.703089\pi$
$463$	8.56075	−	14.8277i	0.397852	−	0.689100i	0.993461	+	0.114175i	$0.0364225\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−25.9318			−1.19998			−0.599990	−	0.800007i	$-0.704829\pi$
$467$	−25.9318			−1.19998			−0.599990	+	0.800007i	$0.704829\pi$
$468$		0			0
$469$	−44.7913			−2.06827
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.172471	−	0.298728i	0.00793020	−	0.0137355i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−10.4574	−	18.1127i	−0.477810	−	0.827591i	0.521867	−	0.853027i	$-0.325236\pi$
$479$	−10.4574	−	18.1127i	−0.477810	−	0.827591i	−0.999676	+	0.0254364i	$0.991902\pi$
$480$		0			0
$481$	4.82424	−	8.35583i	0.219966	−	0.380993i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−20.2373			−0.917039			−0.458520	−	0.888684i	$-0.651620\pi$
$487$	−20.2373			−0.917039			−0.458520	+	0.888684i	$0.651620\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.10009	+	12.2977i	−0.320422	+	0.554988i	−0.980575	−	0.196143i	$-0.937158\pi$
$491$	−7.10009	+	12.2977i	−0.320422	+	0.554988i	0.660153	+	0.751131i	$0.270491\pi$
$492$		0			0
$493$	−21.5681	−	37.3571i	−0.971379	−	1.68248i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.9595	+	38.0350i	0.985018	+	1.70610i
$498$		0			0
$499$	14.1384	−	24.4884i	0.632920	−	1.09625i	−0.354032	−	0.935233i	$-0.615189\pi$
$499$	14.1384	−	24.4884i	0.632920	−	1.09625i	0.986952	−	0.161016i	$-0.0514772\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−17.4165			−0.776565			−0.388282	−	0.921540i	$-0.626931\pi$
$503$	−17.4165			−0.776565			−0.388282	+	0.921540i	$0.626931\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.1060	−	19.2362i	0.492267	−	0.852631i	−0.507694	−	0.861538i	$-0.669502\pi$
$509$	11.1060	−	19.2362i	0.492267	−	0.852631i	0.999960	+	0.00890659i	$0.00283509\pi$
$510$		0			0
$511$	−5.71137	−	9.89238i	−0.252656	−	0.437613i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−7.65116	+	13.2522i	−0.336498	+	0.582831i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−5.48898			−0.240477			−0.120238	−	0.992745i	$-0.538366\pi$
$521$	−5.48898			−0.240477			−0.120238	+	0.992745i	$0.538366\pi$
$522$		0			0
$523$	−26.0991			−1.14123			−0.570617	−	0.821216i	$-0.693296\pi$
$523$	−26.0991			−1.14123			−0.570617	+	0.821216i	$0.693296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.89991	+	3.29075i	−0.0827616	+	0.143347i
$528$		0			0
$529$	−18.8854	−	32.7105i	−0.821105	−	1.42220i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.151158	−	0.261813i	−0.00654737	−	0.0113404i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−71.7529			−3.09062
$540$		0			0
$541$	18.7828			0.807534			0.403767	−	0.914862i	$-0.367701\pi$
$541$	18.7828			0.807534			0.403767	+	0.914862i	$0.367701\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.52825	−	7.84315i	−0.193614	−	0.335349i	0.752831	−	0.658213i	$-0.228688\pi$
$547$	−4.52825	−	7.84315i	−0.193614	−	0.335349i	−0.946445	+	0.322864i	$0.895354\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.95738	−	12.0505i	−0.296395	−	0.513370i
$552$		0			0
$553$	31.5836	−	54.7044i	1.34307	−	2.32627i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−24.8467			−1.05279			−0.526394	−	0.850241i	$-0.676456\pi$
$557$	−24.8467			−1.05279			−0.526394	+	0.850241i	$0.676456\pi$
$558$		0			0
$559$	−0.133217			−0.00563448
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−11.6767	+	20.2247i	−0.492115	+	0.852369i	−0.999959	−	0.00908064i	$-0.997110\pi$
$563$	−11.6767	+	20.2247i	−0.492115	+	0.852369i	0.507843	+	0.861449i	$0.330443\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.54510	−	13.0685i	−0.316307	−	0.547860i	0.663407	−	0.748258i	$-0.269110\pi$
$569$	−7.54510	−	13.0685i	−0.316307	−	0.547860i	−0.979714	+	0.200398i	$0.935776\pi$
$570$		0			0
$571$	3.26786	−	5.66011i	0.136756	−	0.236868i	−0.789511	−	0.613736i	$-0.789666\pi$
$571$	3.26786	−	5.66011i	0.136756	−	0.236868i	0.926267	+	0.376868i	$0.122999\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	8.12662			0.338316			0.169158	−	0.985589i	$-0.445895\pi$
$577$	8.12662			0.338316			0.169158	+	0.985589i	$0.445895\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	34.8743	−	60.4041i	1.44683	−	2.50598i
$582$		0			0
$583$	8.37435	+	14.5048i	0.346830	+	0.600727i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.54937	+	11.3438i	0.270321	+	0.468210i	0.968944	−	0.247280i	$-0.0795367\pi$
$587$	6.54937	+	11.3438i	0.270321	+	0.468210i	−0.698623	+	0.715490i	$0.746203\pi$
$588$		0			0
$589$	−0.612869	+	1.06152i	−0.0252528	+	0.0437392i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.60478			−0.353356			−0.176678	−	0.984269i	$-0.556535\pi$
$593$	−8.60478			−0.353356			−0.176678	+	0.984269i	$0.556535\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.1827	−	21.1010i	0.497771	−	0.862165i	−0.502225	−	0.864737i	$-0.667485\pi$
$599$	12.1827	−	21.1010i	0.497771	−	0.862165i	0.999997	+	0.00257140i	$0.000818504\pi$
$600$		0			0
$601$	12.0900	+	20.9404i	0.493160	+	0.854179i	0.999969	−	0.00787996i	$-0.00250829\pi$
$601$	12.0900	+	20.9404i	0.493160	+	0.854179i	−0.506809	+	0.862059i	$0.669175\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−0.713502	+	1.23582i	−0.0289602	+	0.0501605i	−0.880142	−	0.474710i	$-0.842553\pi$
$607$	−0.713502	+	1.23582i	−0.0289602	+	0.0501605i	0.851182	+	0.524871i	$0.175886\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.90979			0.239085
$612$		0			0
$613$	6.92878			0.279851			0.139925	−	0.990162i	$-0.455314\pi$
$613$	6.92878			0.279851			0.139925	+	0.990162i	$0.455314\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.2466	+	22.9437i	−0.533286	+	0.923679i	0.465958	+	0.884807i	$0.345710\pi$
$617$	−13.2466	+	22.9437i	−0.533286	+	0.923679i	−0.999244	+	0.0388718i	$0.987624\pi$
$618$		0			0
$619$	13.4798	+	23.3478i	0.541801	+	0.938426i	0.998801	+	0.0489597i	$0.0155906\pi$
$619$	13.4798	+	23.3478i	0.541801	+	0.938426i	−0.457000	+	0.889467i	$0.651076\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−25.8722	−	44.8120i	−1.03655	−	1.79535i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−43.7401			−1.74403
$630$		0			0
$631$	−15.9350			−0.634361			−0.317181	−	0.948365i	$-0.602736\pi$
$631$	−15.9350			−0.634361			−0.317181	+	0.948365i	$0.602736\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	13.8556	+	23.9986i	0.548978	+	0.950858i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.58090	−	6.20231i	−0.141437	−	0.244976i	0.786601	−	0.617462i	$-0.211839\pi$
$641$	−3.58090	−	6.20231i	−0.141437	−	0.244976i	−0.928038	+	0.372485i	$0.878506\pi$
$642$		0			0
$643$	−3.53154	+	6.11680i	−0.139270	+	0.241223i	−0.927221	−	0.374516i	$-0.877809\pi$
$643$	−3.53154	+	6.11680i	−0.139270	+	0.241223i	0.787950	+	0.615739i	$0.211142\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−26.4037			−1.03804			−0.519019	−	0.854763i	$-0.673703\pi$
$647$	−26.4037			−1.03804			−0.519019	+	0.854763i	$0.673703\pi$
$648$		0			0
$649$	28.0723			1.10193
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.85985	+	6.68545i	−0.151048	+	0.261622i	−0.931613	−	0.363452i	$-0.881598\pi$
$653$	−3.85985	+	6.68545i	−0.151048	+	0.261622i	0.780565	+	0.625074i	$0.214931\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11.1426	−	19.2996i	−0.434055	−	0.751806i	0.563163	−	0.826346i	$-0.309584\pi$
$659$	−11.1426	−	19.2996i	−0.434055	−	0.751806i	−0.997218	+	0.0745403i	$0.976251\pi$
$660$		0			0
$661$	−16.0544	+	27.8070i	−0.624442	+	1.08157i	0.364207	+	0.931318i	$0.381340\pi$
$661$	−16.0544	+	27.8070i	−0.624442	+	1.08157i	−0.988648	+	0.150247i	$0.951993\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−48.0680			−1.86120
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.81261	+	10.0677i	−0.224393	+	0.388661i
$672$		0			0
$673$	−3.91580	−	6.78237i	−0.150943	−	0.261441i	0.780631	−	0.624992i	$-0.214898\pi$
$673$	−3.91580	−	6.78237i	−0.150943	−	0.261441i	−0.931574	+	0.363551i	$0.881564\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.00171	+	15.5914i	0.345964	+	0.599227i	0.985528	−	0.169511i	$-0.0542187\pi$
$677$	9.00171	+	15.5914i	0.345964	+	0.599227i	−0.639565	+	0.768737i	$0.720885\pi$
$678$		0			0
$679$	23.3269	−	40.4034i	0.895205	−	1.55054i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.3356			1.16076			0.580379	−	0.814347i	$-0.302904\pi$
$683$	30.3356			1.16076			0.580379	+	0.814347i	$0.302904\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.23419	−	5.60179i	0.123213	−	0.213411i
$690$		0			0
$691$	−14.0575	−	24.3482i	−0.534771	−	0.926251i	−0.999174	−	0.0406267i	$-0.987065\pi$
$691$	−14.0575	−	24.3482i	−0.534771	−	0.926251i	0.464403	−	0.885624i	$-0.346269\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.685255	+	1.18690i	−0.0259559	+	0.0449569i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28.4378			−1.07408			−0.537041	−	0.843556i	$-0.680458\pi$
$701$	−28.4378			−1.07408			−0.537041	+	0.843556i	$0.680458\pi$
$702$		0			0
$703$	−14.1096			−0.532153
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.8871	−	34.4455i	0.747932	−	1.29546i
$708$		0			0
$709$	−15.4500	−	26.7602i	−0.580237	−	1.00500i	−0.995451	−	0.0952767i	$-0.969626\pi$
$709$	−15.4500	−	26.7602i	−0.580237	−	1.00500i	0.415213	−	0.909724i	$-0.363707\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.11713	+	3.66698i	0.0792873	+	0.137330i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	21.2462			0.792349			0.396175	−	0.918175i	$-0.370338\pi$
$719$	21.2462			0.792349			0.396175	+	0.918175i	$0.370338\pi$
$720$		0			0
$721$	82.0011			3.05388
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	12.4910	+	21.6351i	0.463267	+	0.802402i	0.999121	−	0.0419082i	$-0.0133437\pi$
$727$	12.4910	+	21.6351i	0.463267	+	0.802402i	−0.535854	+	0.844310i	$0.680010\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.301961	+	0.523013i	0.0111684	+	0.0193443i
$732$		0			0
$733$	−5.23384	+	9.06528i	−0.193316	+	0.334834i	−0.946347	−	0.323151i	$-0.895258\pi$
$733$	−5.23384	+	9.06528i	−0.193316	+	0.334834i	0.753031	+	0.657985i	$0.228591\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	35.8254			1.31964
$738$		0			0
$739$	35.6244			1.31046			0.655232	−	0.755428i	$-0.272571\pi$
$739$	35.6244			1.31046			0.655232	+	0.755428i	$0.272571\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.02770	+	8.70823i	−0.184448	+	0.319474i	−0.943391	−	0.331684i	$-0.892383\pi$
$743$	−5.02770	+	8.70823i	−0.184448	+	0.319474i	0.758942	+	0.651158i	$0.225717\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	28.3914	+	49.1753i	1.03740	+	1.79683i
$750$		0			0
$751$	−8.86740	+	15.3588i	−0.323576	+	0.560450i	−0.981223	−	0.192876i	$-0.938219\pi$
$751$	−8.86740	+	15.3588i	−0.323576	+	0.560450i	0.657647	+	0.753326i	$0.271552\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	6.42042			0.233354			0.116677	−	0.993170i	$-0.462776\pi$
$757$	6.42042			0.233354			0.116677	+	0.993170i	$0.462776\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.6955	−	21.9892i	0.460211	−	0.797108i	−0.538760	−	0.842459i	$-0.681107\pi$
$761$	12.6955	−	21.9892i	0.460211	−	0.797108i	0.998971	+	0.0453506i	$0.0144405\pi$
$762$		0			0
$763$	5.56234	+	9.63425i	0.201370	+	0.348783i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.42080	−	9.38910i	−0.195734	−	0.339021i
$768$		0			0
$769$	7.40088	−	12.8187i	0.266883	−	0.462254i	−0.701173	−	0.712992i	$-0.747340\pi$
$769$	7.40088	−	12.8187i	0.266883	−	0.462254i	0.968055	+	0.250737i	$0.0806731\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.48046			0.269053			0.134527	−	0.990910i	$-0.457049\pi$
$773$	7.48046			0.269053			0.134527	+	0.990910i	$0.457049\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.221048	+	0.382866i	−0.00791985	+	0.0137176i
$780$		0			0
$781$	−17.5639	−	30.4215i	−0.628484	−	1.08857i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−26.1587	+	45.3082i	−0.932457	+	1.61506i	−0.153350	+	0.988172i	$0.549006\pi$
$787$	−26.1587	+	45.3082i	−0.932457	+	1.61506i	−0.779107	+	0.626891i	$0.784327\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.1575			−0.503384
$792$		0			0
$793$	4.48969			0.159434
$794$		0			0
$795$		0			0
$796$		0			0
$797$	26.8679	−	46.5366i	0.951711	−	1.64841i	0.209991	−	0.977703i	$-0.432657\pi$
$797$	26.8679	−	46.5366i	0.951711	−	1.64841i	0.741721	−	0.670709i	$-0.234010\pi$
$798$		0			0
$799$	−13.3957	−	23.2019i	−0.473904	−	0.820826i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.56812	+	7.91222i	0.161205	+	0.279216i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.1908			0.920819			0.460410	−	0.887707i	$-0.347703\pi$
$809$	26.1908			0.920819			0.460410	+	0.887707i	$0.347703\pi$
$810$		0			0
$811$	−52.3506			−1.83828			−0.919140	−	0.393931i	$-0.871115\pi$
$811$	−52.3506			−1.83828			−0.919140	+	0.393931i	$0.871115\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.0974059	+	0.168712i	0.00340780	+	0.00590249i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.42932	−	4.20771i	−0.0847839	−	0.146850i	0.820515	−	0.571625i	$-0.193687\pi$
$821$	−2.42932	−	4.20771i	−0.0847839	−	0.146850i	−0.905299	+	0.424775i	$0.860353\pi$
$822$		0			0
$823$	0.588886	−	1.01998i	0.0205273	−	0.0355543i	−0.855579	−	0.517672i	$-0.826799\pi$
$823$	0.588886	−	1.01998i	0.0205273	−	0.0355543i	0.876107	+	0.482117i	$0.160132\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	32.7188			1.13775			0.568873	−	0.822426i	$-0.307380\pi$
$827$	32.7188			1.13775			0.568873	+	0.822426i	$0.307380\pi$
$828$		0			0
$829$	−5.99342			−0.208160			−0.104080	−	0.994569i	$-0.533190\pi$
$829$	−5.99342			−0.208160			−0.104080	+	0.994569i	$0.533190\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	62.8125	−	108.794i	2.17632	−	3.76951i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19.1933	−	33.2439i	−0.662628	−	1.14771i	−0.979923	−	0.199378i	$-0.936108\pi$
$839$	−19.1933	−	33.2439i	−0.662628	−	1.14771i	0.317295	−	0.948327i	$-0.397226\pi$
$840$		0			0
$841$	−4.51023	+	7.81195i	−0.155525	+	0.269378i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	24.8083			0.852425
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.3705	+	42.2109i	−0.835410	+	1.44697i
$852$		0			0
$853$	10.8595	+	18.8092i	0.371822	+	0.644015i	0.989846	−	0.142145i	$-0.0453998\pi$
$853$	10.8595	+	18.8092i	0.371822	+	0.644015i	−0.618024	+	0.786159i	$0.712067\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2.29983	+	3.98342i	0.0785607	+	0.136071i	0.902629	−	0.430419i	$-0.141634\pi$
$857$	2.29983	+	3.98342i	0.0785607	+	0.136071i	−0.824068	+	0.566490i	$0.808301\pi$
$858$		0			0
$859$	−7.45896	+	12.9193i	−0.254496	+	0.440801i	−0.964759	−	0.263137i	$-0.915243\pi$
$859$	−7.45896	+	12.9193i	−0.254496	+	0.440801i	0.710262	+	0.703937i	$0.248576\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	11.4089			0.388362			0.194181	−	0.980966i	$-0.437795\pi$
$863$	11.4089			0.388362			0.194181	+	0.980966i	$0.437795\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25.2615	+	43.7541i	−0.856937	+	1.48426i
$870$		0			0
$871$	−6.91793	−	11.9822i	−0.234405	−	0.406001i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	13.9642	−	24.1867i	0.471537	−	0.816727i	−0.527932	−	0.849286i	$-0.677033\pi$
$877$	13.9642	−	24.1867i	0.471537	−	0.816727i	0.999470	+	0.0325596i	$0.0103659\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	42.3270			1.42603			0.713017	−	0.701147i	$-0.247328\pi$
$881$	42.3270			1.42603			0.713017	+	0.701147i	$0.247328\pi$
$882$		0			0
$883$	20.3895			0.686161			0.343081	−	0.939306i	$-0.388530\pi$
$883$	20.3895			0.686161			0.343081	+	0.939306i	$0.388530\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−19.7725	+	34.2471i	−0.663897	+	1.14990i	0.315686	+	0.948864i	$0.397765\pi$
$887$	−19.7725	+	34.2471i	−0.663897	+	1.14990i	−0.979583	+	0.201040i	$0.935568\pi$
$888$		0			0
$889$	17.5383	+	30.3772i	0.588215	+	1.01882i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.32113	−	7.48442i	−0.144601	−	0.250457i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.34919			0.111702
$900$		0			0
$901$	−29.3236			−0.976911
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	18.4798	+	32.0080i	0.613613	+	1.06281i	0.990626	+	0.136601i	$0.0436179\pi$
$907$	18.4798	+	32.0080i	0.613613	+	1.06281i	−0.377013	+	0.926208i	$0.623049\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.92335	−	8.52749i	−0.163118	−	0.282528i	0.772867	−	0.634568i	$-0.218822\pi$
$911$	−4.92335	−	8.52749i	−0.163118	−	0.282528i	−0.935985	+	0.352039i	$0.885488\pi$
$912$		0			0
$913$	−27.8935	+	48.3130i	−0.923140	+	1.59893i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	41.7955			1.38021
$918$		0			0
$919$	−8.42079			−0.277776			−0.138888	−	0.990308i	$-0.544353\pi$
$919$	−8.42079			−0.277776			−0.138888	+	0.990308i	$0.544353\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−6.78321	+	11.7489i	−0.223272	+	0.386718i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14.1938	−	24.5843i	−0.465683	−	0.806586i	0.533549	−	0.845769i	$-0.320858\pi$
$929$	−14.1938	−	24.5843i	−0.465683	−	0.806586i	−0.999232	+	0.0391829i	$0.987524\pi$
$930$		0			0
$931$	20.2619	−	35.0946i	0.664057	−	1.15018i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.69342			0.153327			0.0766637	−	0.997057i	$-0.475573\pi$
$937$	4.69342			0.153327			0.0766637	+	0.997057i	$0.475573\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.51278	−	16.4766i	0.310108	−	0.537123i	−0.668278	−	0.743912i	$-0.732968\pi$
$941$	9.51278	−	16.4766i	0.310108	−	0.537123i	0.978385	+	0.206789i	$0.0663015\pi$
$942$		0			0
$943$	0.763600	+	1.32259i	0.0248662	+	0.0430696i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.40843	+	11.0997i	0.208246	+	0.360693i	0.951162	−	0.308692i	$-0.0998912\pi$
$947$	6.40843	+	11.0997i	0.208246	+	0.360693i	−0.742916	+	0.669385i	$0.766558\pi$
$948$		0			0
$949$	1.76422	−	3.05572i	0.0572690	−	0.0991928i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−45.9701			−1.48912			−0.744560	−	0.667556i	$-0.767340\pi$
$953$	−45.9701			−1.48912			−0.744560	+	0.667556i	$0.767340\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.97230	+	12.0764i	−0.225147	+	0.389967i
$960$		0			0
$961$	15.3525	+	26.5913i	0.495242	+	0.857783i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.57354	−	4.45751i	0.0827596	−	0.143344i	−0.821675	−	0.569957i	$-0.806960\pi$
$967$	2.57354	−	4.45751i	0.0827596	−	0.143344i	0.904434	+	0.426613i	$0.140293\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10.3875			0.333352			0.166676	−	0.986012i	$-0.446697\pi$
$971$	10.3875			0.333352			0.166676	+	0.986012i	$0.446697\pi$
$972$		0			0
$973$	−19.5632			−0.627169
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.7811	+	34.2618i	−0.632852	+	1.09613i	0.354113	+	0.935202i	$0.384783\pi$
$977$	−19.7811	+	34.2618i	−0.632852	+	1.09613i	−0.986966	+	0.160930i	$0.948551\pi$
$978$		0			0
$979$	20.6933	+	35.8419i	0.661362	+	1.14551i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−10.2360	−	17.7292i	−0.326477	−	0.565474i	0.655333	−	0.755340i	$-0.272528\pi$
$983$	−10.2360	−	17.7292i	−0.326477	−	0.565474i	−0.981810	+	0.189865i	$0.939195\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.672970			0.0213992
$990$		0			0
$991$	−19.2415			−0.611228			−0.305614	−	0.952156i	$-0.598862\pi$
$991$	−19.2415			−0.611228			−0.305614	+	0.952156i	$0.598862\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.6193	−	20.1251i	−0.367986	−	0.637370i	0.621265	−	0.783601i	$-0.286619\pi$
$997$	−11.6193	−	20.1251i	−0.367986	−	0.637370i	−0.989250	+	0.146231i	$0.953286\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.i.d.901.1		8
3.2	odd	2		900.2.i.e.301.3	yes	8
5.2	odd	4		2700.2.s.d.1549.8		16
5.3	odd	4		2700.2.s.d.1549.1		16
5.4	even	2		2700.2.i.e.901.4		8
9.2	odd	6		900.2.i.e.601.3	yes	8
9.4	even	3		8100.2.a.ba.1.4		4
9.5	odd	6		8100.2.a.z.1.4		4
9.7	even	3	inner	2700.2.i.d.1801.1		8
15.2	even	4		900.2.s.d.49.7		16
15.8	even	4		900.2.s.d.49.2		16
15.14	odd	2		900.2.i.d.301.2	✓	8
45.2	even	12		900.2.s.d.349.2		16
45.4	even	6		8100.2.a.y.1.1		4
45.7	odd	12		2700.2.s.d.2449.1		16
45.13	odd	12		8100.2.d.s.649.1		8
45.14	odd	6		8100.2.a.x.1.1		4
45.22	odd	12		8100.2.d.s.649.8		8
45.23	even	12		8100.2.d.q.649.1		8
45.29	odd	6		900.2.i.d.601.2	yes	8
45.32	even	12		8100.2.d.q.649.8		8
45.34	even	6		2700.2.i.e.1801.4		8
45.38	even	12		900.2.s.d.349.7		16
45.43	odd	12		2700.2.s.d.2449.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.2	✓	8	15.14	odd	2
900.2.i.d.601.2	yes	8	45.29	odd	6
900.2.i.e.301.3	yes	8	3.2	odd	2
900.2.i.e.601.3	yes	8	9.2	odd	6
900.2.s.d.49.2		16	15.8	even	4
900.2.s.d.49.7		16	15.2	even	4
900.2.s.d.349.2		16	45.2	even	12
900.2.s.d.349.7		16	45.38	even	12
2700.2.i.d.901.1		8	1.1	even	1	trivial
2700.2.i.d.1801.1		8	9.7	even	3	inner
2700.2.i.e.901.4		8	5.4	even	2
2700.2.i.e.1801.4		8	45.34	even	6
2700.2.s.d.1549.1		16	5.3	odd	4
2700.2.s.d.1549.8		16	5.2	odd	4
2700.2.s.d.2449.1		16	45.7	odd	12
2700.2.s.d.2449.8		16	45.43	odd	12
8100.2.a.x.1.1		4	45.14	odd	6
8100.2.a.y.1.1		4	45.4	even	6
8100.2.a.z.1.4		4	9.5	odd	6
8100.2.a.ba.1.4		4	9.4	even	3
8100.2.d.q.649.1		8	45.23	even	12
8100.2.d.q.649.8		8	45.32	even	12
8100.2.d.s.649.1		8	45.13	odd	12
8100.2.d.s.649.8		8	45.22	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.49787 - 4.32643i) q^{7} +(1.99787 + 3.46041i) q^{11} +(0.771582 - 1.33642i) q^{13} -6.99574 q^{17} -2.25667 q^{19} +(-3.89778 + 6.75116i) q^{23} +(3.08304 + 5.33998i) q^{29} +(0.271582 - 0.470394i) q^{31} +6.25240 q^{37} +(0.0979532 - 0.169660i) q^{41} +(-0.0431636 - 0.0747616i) q^{43} +(1.91483 + 3.31658i) q^{47} +(-8.97869 + 15.5515i) q^{49} +4.19164 q^{53} +(3.51278 - 6.08432i) q^{59} +(1.45470 + 2.51962i) q^{61} +(4.48295 - 7.76470i) q^{67} -8.79130 q^{71} +2.28650 q^{73} +(9.98082 - 17.2873i) q^{77} +(6.32211 + 10.9502i) q^{79} +(6.98082 + 12.0911i) q^{83} +10.3577 q^{89} -7.70924 q^{91} +(4.66936 + 8.08758i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{7} - 3 q^{11} + 2 q^{13} - 18 q^{17} - 8 q^{19} - 3 q^{23} + 9 q^{29} - 2 q^{31} + 2 q^{37} - 9 q^{41} + 8 q^{43} + 12 q^{47} - 9 q^{49} - 24 q^{53} + 15 q^{59} + q^{61} + 11 q^{67} + 24 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) 8 * q - q^7 - 3 * q^11 + 2 * q^13 - 18 * q^17 - 8 * q^19 - 3 * q^23 + 9 * q^29 - 2 * q^31 + 2 * q^37 - 9 * q^41 + 8 * q^43 + 12 * q^47 - 9 * q^49 - 24 * q^53 + 15 * q^59 + q^61 + 11 * q^67 + 24 * q^71 + 20 * q^73 + 36 * q^77 + 7 * q^79 + 12 * q^83 - 6 * q^89 - 22 * q^91 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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